#378675

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=f,d^2=efh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di

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, 285871);