#7611068

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1426 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,20,26)(21,27,38,46)(22,28,23,29)(31,39,33,41)(32,40,34,42)(35,43,47,48)(36,44,37,45);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,24,20)(18,25,30,26)(21,35,38,47)(22,32,23,34)(27,43,46,48)(28,40,29,42)(31,36,33,37)(39,44,41,45);(10,14)(11,15)(17,20,24,19)(18,26,30,25)(21,31,38,33)(22,32,23,34)(27,39,46,41)(28,40,29,42)(35,36,47,37)(43,44,48,45);(10,14)(13,16)(17,21,22,36,24,38,23,37)(18,27,28,44,30,46,29,45)(19,31,34,47,20,33,32,35)(25,39,42,48,26,41,40,43);(1,3,4,8)(2,6,5,7)(17,22,24,23)(18,28,30,29)(19,32,20,34)(21,36,38,37)(25,40,26,42)(27,44,46,45)(31,35,33,47)(39,43,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518264);