#6370976

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4166 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278172);