#6466134

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2242 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)(7,11)(8,9)(10,12)(13,14,19,34,18,23,17,22)(15,24,27,20,26,37,28,36)(16,29,31,40,33,21,32,39)(25,42,44,38,30,41,43,35);(2,3)(6,9)(10,11)(13,15)(14,20)(16,30)(17,27)(18,26)(19,28)(21,38)(22,37)(23,36)(24,34)(25,33)(29,35)(31,44)(32,43)(39,42)(40,41);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,25)(17,32)(18,33)(19,31)(20,35)(22,40)(23,29)(24,41)(26,30)(27,44)(28,43)(34,39)(36,38)(37,42);(1,3,4,2)(15,26)(20,36)(24,37)(25,30)(27,28)(35,38)(41,42)(43,44);(13,17,18,19)(14,22,23,34)(15,27,26,28)(16,31,33,32)(20,37,36,24)(21,39,29,40)(25,43,30,44)(35,41,38,42)

Polycyclic presentation:

2^9|a^2=fg,c^2=gh,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, 6373330);