#188686

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=h,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95882);