#361069

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,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, 268265);