#6463014

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×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

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,e^a=ei,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, 6370210);