#7027532

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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1604 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)(20,31,34,38)(21,25,36,42)(24,32,40,28)(29,41,35,37)(33,44,46,47)(39,45,48,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,35,36,29)(24,38,40,31)(25,41,42,37)(33,43,46,45)(39,44,48,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,28,22,34,30,32)(18,24,23,31,26,40,27,38)(21,33,29,43,36,46,35,45)(25,39,37,44,42,48,41,47);(2,6)(7,8)(11,16)(13,15)(17,21)(18,25)(19,29)(20,33)(22,36)(23,37)(24,39)(26,42)(27,41)(28,43)(30,35)(31,44)(32,45)(34,46)(38,47)(40,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,26)(20,34)(21,36)(23,27)(28,32)(29,35)(39,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,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=ehi,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, 6934728);