#360489

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267685);