#6767993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(4,6)(5,8)(17,18,23,27)(19,28,29,42)(20,25,33,39)(21,26,22,36)(24,31,37,32)(30,43,44,48)(34,40,35,41)(38,46,47,45);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(19,29)(20,33)(24,37)(25,39)(28,42)(31,32)(34,35)(40,41);(10,13)(14,16)(17,19,23,29)(18,24,27,37)(20,30,33,44)(21,32,22,31)(25,38,39,47)(26,42,36,28)(34,46,35,45)(40,43,41,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,23,33)(18,25,27,39)(19,30,29,44)(21,35,22,34)(24,38,37,47)(26,41,36,40)(28,43,42,48)(31,46,32,45);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,34,33,35)(24,28,37,42)(25,40,39,41)(30,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,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, 6675189);