#7594821

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	#44976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 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)(17,18,22,27)(19,26,29,25)(20,31,30,32)(21,24,28,23);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,25,27,26)(20,24,30,23)(21,32,28,31);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,23,19,22,30,24,29)(18,26,28,31,27,25,21,32);(9,11,12,16)(10,14,13,15)(17,21,23,18,22,28,24,27)(19,26,30,31,29,25,20,32);(2,5)(6,8)(18,27)(21,28)(25,26)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=g,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, 7502017);