#368896

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei

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, 276092);