#6467415

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2245 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6374611);