#386026

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1630 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)(5,6,7,8)(9,10)(11,22)(12,17)(13,29)(14,20)(15,21)(16,25)(18,28)(19,32)(23,38)(24,40)(26,34)(27,33)(30,37)(31,35)(36,39);(9,11)(10,16)(12,27)(13,23)(14,31)(15,26)(17,36)(18,38)(19,34)(20,22)(21,35)(24,28)(25,32)(29,33)(30,39)(37,40);(2,4)(9,12,14,13,15,28,32,30)(10,17,20,29,21,18,19,37)(11,23,25,27,26,39,31,24)(16,33,34,36,35,40,22,38);(2,4)(9,13,14,28,15,30,32,12)(10,18,19,29,21,17,20,37)(11,24,25,23,26,27,31,39)(16,33,22,38,35,40,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293222);