#7135102

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(9,10,14,11)(12,15,13,16)(17,18,23,25,21,27,22,28)(19,29,35,39,31,24,32,37)(20,26,36,40,33,41,34,42)(30,43,46,48,44,38,45,47);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,21,31)(18,24,27,29)(20,30,33,44)(22,32,23,35)(25,37,28,39)(26,38,41,43)(34,45,36,46)(40,47,42,48);(9,11,14,10)(12,15,13,16)(18,25)(22,23)(24,37)(26,40)(27,28)(29,39)(32,35)(34,36)(38,47)(41,42)(43,48)(45,46);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20,21,33)(18,26,27,41)(19,30,31,44)(22,34,23,36)(24,38,29,43)(25,40,28,42)(32,45,35,46)(37,47,39,48);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(20,33)(26,41)(30,44)(34,36)(38,43)(40,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7042298);