#121927

order	512
nilpotency class	3
isoclinism	isoclinic class of #121925
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

344 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,d^c=dh,e^c=eh,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, 29123);