#405485

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312681);