#384219

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1047 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)(8,10)(9,12)(13,14)(15,26)(16,30)(17,22)(18,35)(19,25)(20,24)(21,28)(23,32)(27,43)(29,40)(31,37)(33,42)(34,41)(36,39)(38,44);(13,15,18,28,19,29,20,36)(14,21,24,26,25,39,35,40)(16,31,32,27,17,33,34,44)(22,38,23,42,30,43,41,37);(5,7,8,12)(6,9,10,11)(13,16,19,17)(14,22,25,30)(15,27,29,44)(18,32,20,34)(21,37,39,42)(23,35,41,24)(26,38,40,43)(28,33,36,31);(6,10)(9,11)(13,17,19,16)(14,23,25,41)(15,27,29,44)(18,32,20,34)(21,38,39,43)(22,35,30,24)(26,37,40,42)(28,31,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291415);