#7248130

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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

1442 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7155326);