#495883

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33604	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(3,4)(5,6,10,15,11,17,25,16)(7,18,23,33,20,29,36,34)(8,13,9,24,19,14,26,32)(12,27,30,35,28,21,31,22);(5,7,19,27,10,23,26,35,11,20,8,21,25,36,9,22)(6,12,24,34,16,31,13,29,17,28,32,33,15,30,14,18);(2,4)(7,20)(12,28)(18,29)(21,27)(22,35)(23,36)(30,31)(33,34);(5,8,10,9,11,19,25,26)(6,13,15,24,17,14,16,32)(7,21,23,22,20,27,36,35)(12,29,30,34,28,18,31,33)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403079);