#7000152

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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=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, 6907348);