#7596448

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

1488 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,29,21,25)(20,22,28,24)(26,30,31,32);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,27,29)(20,31,28,26)(22,30,24,32);(10,13)(14,15)(17,20,24,18,23,28,22,27)(19,25,30,31,21,29,32,26);(9,11,12,16)(10,14,13,15)(17,21,24,32,23,19,22,30)(18,26,28,25,27,31,20,29);(1,3,4,8)(2,6,5,7)(17,22,23,24)(18,20,27,28)(19,30,21,32)(25,31,29,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=f,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, 7503644);