#7027406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6934602);