#7597201

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504397);