#360483

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

1050 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:

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

Polycyclic presentation:

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

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, 267679);