#185188

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2482 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)(5,6,9,12,10,13,8,17)(7,14,15,19,16,11,18,20)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29);(2,4);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92384);