#7620704

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	#24267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1667 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);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(19,25)(20,27)(22,29)(24,30)(26,31)(28,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,18,21,23)(19,26,28,22)(20,24,27,30)(25,31,32,29);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,19,27,32,21,28,20,25)(18,22,30,31,23,26,24,29);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(18,23)(22,26)(24,30)(29,31)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=fghi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,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.

Relationships

maximal subgroup of	#72313233, #72313405, #72313579, …
maximal normal subgroup of	#72313233, #72313405, #72313579, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527900);