#7620712

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#24277	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18)(19,24)(20,28)(21,23)(22,26)(25,31)(27,29)(30,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(18,23)(20,29)(22,24)(25,27)(28,30)(31,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19,21,26)(18,24,23,22)(20,30,31,27)(25,29,28,32);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,24,32,21,31,22,29)(18,25,19,27,23,28,26,30);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(18,23)(19,26)(25,28)(27,30)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fgh,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527908);