#7257412

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1566 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)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,31)(24,34)(27,35)(29,39)(30,42)(32,40)(33,43)(37,41)(38,47)(44,46)(45,48);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,29)(21,31)(22,32)(24,38)(25,42)(26,28)(27,40)(30,33)(34,46)(35,48)(36,45)(37,47)(39,43)(41,44);(10,13)(11,16)(17,20,21,33)(18,24,26,41)(19,29,31,30)(22,37,35,34)(23,38,28,44)(25,27,43,36)(32,47,48,46)(39,45,42,40);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,39)(24,36)(26,43)(27,41)(28,42)(29,31)(32,47)(35,37)(38,40)(44,45)(46,48);(2,5)(6,8)(10,13)(14,15)(18,26)(19,31)(20,33)(27,36)(32,48)(34,37)(38,44)(39,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7164608);