#6464025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1838 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6371221);