#7594823

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,19)(20,22,26,27)(21,28,30,29)(24,32,25,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,27,26,22)(21,29,30,28)(24,31,25,32);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,19,23,26,27,18)(21,28,31,25,30,29,32,24);(9,11,12,16)(10,14,13,15)(17,21,27,32,23,30,22,31)(18,24,26,29,19,25,20,28);(2,5)(6,8)(21,30)(24,25)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=gi,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, 7502019);