#6463020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2290 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)(13,14,19,23,18,22,17,24)(15,25,28,20,27,40,29,35)(16,21,31,39,33,37,32,38)(26,41,43,34,30,44,42,36);(7,8)(10,12)(13,15)(14,20)(16,30)(17,28)(18,27)(19,29)(21,36)(22,35)(23,25)(24,40)(26,33)(31,43)(32,42)(34,37)(38,41)(39,44);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,31)(20,34)(22,37)(23,39)(24,38)(25,41)(27,30)(28,43)(29,42)(35,36)(40,44);(5,8,9,7)(6,10,11,12)(14,22)(15,27)(21,37)(23,24)(26,30)(28,29)(38,39)(42,43);(13,17,18,19)(14,23,22,24)(15,28,27,29)(16,31,33,32)(20,25,35,40)(21,38,37,39)(26,42,30,43)(34,44,36,41)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370216);