#7195002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 7102198);