#7010564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,36)(19,24,31,40,32,41,30,47)(20,26,34,44,35,43,37,25)(29,39,46,42,33,45,48,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,41)(20,33,35,29)(21,30,22,31)(25,42,44,38)(26,45,43,39)(27,40,36,47)(34,48,37,46);(9,11)(10,13)(12,16)(14,15)(17,20,23,35)(18,25,28,44)(19,29,32,33)(21,34,22,37)(24,38,41,42)(26,27,43,36)(30,46,31,48)(39,40,45,47);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,29)(21,37)(22,34)(23,35)(24,39)(25,36)(27,44)(28,43)(30,48)(31,46)(32,33)(38,47)(40,42)(41,45);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,27,28,36)(19,30,32,31)(20,34,35,37)(24,40,41,47)(25,43,44,26)(29,46,33,48)(38,45,42,39)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6917760);