#7156989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45355	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2220 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292956, #72292967, #72293001, …
maximal normal subgroup of	#72292956, #72292967, #72293001, …
maximal quotient of	C₂×#7156989

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7064185);