#495676

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3002	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2171, #72343063
maximal normal subgroup of	D₄×#2171
maximal quotient of	#72287273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402872);