#212188

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=dh,d^c=di,f^c=fh

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, 119384);