#430001

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1576 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337197);