#6930056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#39185	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,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, 6837252);