#7594848

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,26,33,25)(20,34,32,42)(21,36,39,45)(22,30,24,28)(27,38,46,40)(31,47,48,44)(35,41,37,43);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,24)(21,37)(22,32)(23,33)(26,29)(27,44)(28,42)(30,34)(31,40)(35,39)(36,41)(38,48)(43,45)(46,47);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,33,23,32,24,19)(18,26,28,42,29,25,30,34)(21,35,38,48,39,37,40,31)(27,43,45,44,46,41,36,47);(9,11,12,16)(10,14,13,15)(17,21,22,38,23,39,24,40)(18,27,28,45,29,46,30,36)(19,31,20,35,33,48,32,37)(25,41,34,47,26,43,42,44);(2,5)(6,8)(17,22,23,24)(18,28,29,30)(19,32,33,20)(21,38,39,40)(25,42,26,34)(27,45,46,36)(31,37,48,35)(41,44,43,47)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=gi,e^2=i,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^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502044);