#7135035

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(17,18,21,26,22,27,33,24)(19,23,30,37,29,36,43,34)(20,25,32,40,31,39,44,38)(28,35,41,46,42,47,48,45);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,22,29)(18,23,27,36)(20,28,31,42)(21,30,33,43)(24,34,26,37)(25,35,39,47)(32,41,44,48)(38,45,40,46);(9,11,14,10)(12,15,13,16)(18,24)(21,33)(23,34)(25,38)(26,27)(30,43)(32,44)(35,45)(36,37)(39,40)(41,48)(46,47);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(29,42)(30,41)(33,44)(34,45)(36,47)(37,46)(43,48);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(19,29)(20,31)(23,36)(25,39)(30,43)(32,44)(34,37)(38,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 7042231);