#142247

order	512
nilpotency class	4
isoclinism	isoclinic class of #142246
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

715 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,28,53)(12,29,33,46)(13,34,38,49)(14,21,15,20)(17,40,43,25)(18,39,47,32)(19,36,50,37)(24,54,57,71)(26,59,27,48)(30,62,56,52)(31,64,67,45)(35,51,58,42)(41,63,60,72)(44,65,69,55)(61,68,70,66);(5,8)(9,11)(10,17)(12,30)(13,35)(14,26)(15,27)(16,28)(18,44)(19,48)(20,42)(21,51)(22,43)(23,34)(24,32)(25,37)(29,60)(31,65)(33,56)(36,40)(38,58)(39,57)(41,46)(45,52)(47,69)(49,53)(50,59)(54,61)(55,67)(62,64)(63,66)(68,72)(70,71);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,24)(13,31)(14,39)(15,32)(16,33)(17,41)(19,45)(20,46)(21,29)(22,47)(23,52)(25,55)(26,30)(27,56)(28,57)(34,61)(35,63)(36,68)(37,66)(38,67)(40,65)(42,69)(43,60)(44,51)(48,71)(49,70)(50,64)(53,62)(54,59)(58,72)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49443);