#6412265

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#36684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,34,42)(19,23,30,36,29,25,40,41)(20,31,33,45,32,24,37,39)(28,38,44,48,43,35,46,47);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,26)(27,36)(31,38)(32,43)(33,44)(34,40)(37,46)(39,47)(41,42)(45,48);(10,14)(11,15)(17,20,22,33,21,32,34,37)(18,24,27,39,26,31,42,45)(19,28,30,44,29,43,40,46)(23,35,36,47,25,38,41,48);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,38)(27,41)(31,35)(32,43)(33,44)(34,40)(36,42)(37,46)(39,48)(45,47);(1,5,6,4)(2,7,3,8)(19,29)(23,25)(28,43)(30,40)(35,38)(36,41)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6319461);