#367499

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

1195 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^c=gi,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, 274695);