#185230

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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	C₂×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92426);