#7090754

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
prime radical	#23848	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,20,23,26,21,24)(19,25,29,30,27,32,31,28);(9,10)(11,15)(12,14)(13,16)(18,24)(19,27)(20,26)(21,22)(25,30)(28,32);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,27)(18,25,26,32)(20,30,24,28)(21,31,22,29);(11,16)(13,15)(17,20,21,18,23,24,22,26)(19,28,31,32,27,30,29,25);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,20,26,24)(19,29,27,31)(25,28,32,30)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,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	#72251190, #72251222, #72251266, …
maximal normal subgroup of	#72251190, #72251222, #72251266, …
maximal quotient of	#72264718

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997950);