#6911125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(9,10,12,13)(11,15,16,14)(17,18,22,24)(19,25,27,20)(21,29,31,23)(26,28,32,30);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,20,24,25)(21,26,31,32)(23,28,29,30);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,20,19,24,22,25,27,18)(21,30,26,23,31,28,32,29);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,31)(18,23,24,29)(19,26,27,32)(20,28,25,30);(2,6)(5,7)(21,31)(23,29)(26,32)(28,30)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308665, #72308819
maximal normal subgroup of	#72308665, #72308819

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6818321);