#7052327

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1469 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,e^2=f,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,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6959523);