#7038638

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

1393 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)(10,13)(11,15)(17,18,21,26)(19,28,30,23)(20,25,33,42)(22,24,35,27)(29,40,32,37)(31,38,44,36)(34,39,46,41)(43,48,45,47);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(17,19)(18,23)(20,32)(21,30)(22,31)(24,36)(25,40)(26,28)(27,38)(29,33)(34,45)(35,44)(37,42)(39,48)(41,47)(43,46);(9,10)(11,14)(12,13)(15,16)(18,24)(20,33)(22,35)(23,36)(25,41)(26,27)(28,38)(29,32)(31,44)(37,48)(39,42)(40,47);(3,8)(6,7)(10,13)(14,16)(17,20,21,33)(18,25,26,42)(19,29,30,32)(22,34,35,46)(23,37,28,40)(24,39,27,41)(31,43,44,45)(36,47,38,48);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(19,30)(20,33)(23,28)(25,42)(31,44)(34,46)(36,38)(39,41)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,d^2=i,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=dhi,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, 6945834);