#374690

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=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, 281886);