#368883

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1637 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)(5,6,11,15,12,16,10,8)(7,17,20,13,21,28,25,18)(9,22,23,14,24,30,26,29)(19,27,34,36,35,32,33,31);(1,2)(3,4)(5,7,11,20,12,21,10,25)(6,13,15,28,16,18,8,17)(9,19,23,34,24,35,26,33)(14,27,30,36,29,32,22,31);(2,4)(5,8,11,16,12,15,10,6)(7,18,20,28,21,13,25,17)(9,22,23,29,24,30,26,14)(19,32,34,36,35,27,33,31);(5,9,12,24)(6,14,16,29)(7,19,21,35)(8,22,15,30)(10,26,11,23)(13,27,18,32)(17,31,28,36)(20,34,25,33)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=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, 276079);