#498112

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

750 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)(5,6,10,14,9,15,26,19,11,16,27,7,17,13,31,20)(8,12,23,29,24,28,35,34,25,30,36,18,22,32,33,21);(5,7,10,13,9,20,26,6,11,14,27,15,17,19,31,16)(8,21,23,12,24,29,35,28,25,34,36,30,22,18,33,32);(2,4);(5,8)(6,12)(7,18)(9,24)(10,23)(11,25)(13,32)(14,29)(15,28)(16,30)(17,22)(19,34)(20,21)(26,35)(27,36)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2171, #72343077
maximal normal subgroup of	D₄×#2171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405308);