#7154346

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7061542);