#368963

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=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, 276159);