#144401

order	512
nilpotency class	4
isoclinism	isoclinic class of #144401
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10337
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#23899	C₂
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,f^d=fi,f^e=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51597);