#7195088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281810, #72282141, #72285261
maximal normal subgroup of	#72281810, #72282141, #72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102284);