#6463002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370198);