#6382812

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290008);