#7004030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6911226);