#493102

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

848 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)(7,19,22,32)(8,14)(9,16)(10,17)(11,18)(12,15)(13,21,30,26)(20,23,33,35)(24,27)(25,28)(29,31,34,36);(5,7,10,23,11,21,27,34,12,22,28,35,9,26,14,29)(6,13,17,31,18,19,24,33,15,30,25,36,16,32,8,20);(1,3,4,2)(5,8,9,25,12,24,11,17)(6,14,16,28,15,27,18,10)(7,20,26,36,22,33,21,31)(13,29,32,35,30,34,19,23);(6,15)(8,24)(13,30)(16,18)(17,25)(19,32)(20,33)(31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400298);