#7224401

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1260 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)(4,8)(5,7)(11,15)(14,16)(17,18,22,28)(19,30,34,24)(20,35,37,26)(21,29,23,27)(25,39,44,36)(31,46,32,41)(33,42,38,45)(40,48,43,47);(1,3,6,2)(4,8,5,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,24)(20,32)(21,38)(22,34)(23,33)(25,43)(26,41)(27,45)(28,30)(29,42)(31,37)(35,46)(36,47)(39,48)(40,44);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,25)(19,31)(21,36)(22,37)(23,39)(24,40)(26,29)(27,35)(28,44)(30,43)(32,34)(33,47)(38,48)(41,45)(42,46);(1,4,6,5)(2,8,3,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,32)(21,39)(22,37)(23,36)(24,41)(25,29)(27,44)(28,35)(30,46)(31,34)(33,47)(38,48)(40,45)(42,43);(1,5,6,4)(2,8,3,7)(10,13)(14,16)(17,21,22,23)(18,27,28,29)(19,33,34,38)(20,36,37,39)(24,42,30,45)(25,35,44,26)(31,47,32,48)(40,46,43,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7131597);