#379293

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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

945 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)(7,10)(8,9)(11,12)(13,14,17,22,19,24,18,23)(15,25,28,20,27,35,29,36)(16,30,31,38,33,37,32,21)(26,34,41,43,40,44,42,39);(1,2,3,4)(6,9)(10,12)(13,15,18,29,19,27,17,28)(14,20,22,35,24,36,23,25)(16,26,32,42,33,40,31,41)(21,34,30,43,38,44,37,39);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,31,19,33,18,32)(14,21,23,37,24,38,22,30)(15,26,28,41,27,40,29,42)(20,34,25,39,36,44,35,43);(6,9)(10,12)(14,22,24,23)(15,27)(20,25,36,35)(21,30,38,37)(26,40)(28,29)(34,39,44,43)(41,42)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=e,d^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 286489);