#464079

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 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)(5,6,12,8)(7,17,22,18)(9,14,27,24)(10,16,11,15)(13,21,23,20)(19,32,35,31)(25,28,30,26)(29,36,33,34);(5,7,11,21,12,22,10,20)(6,13,16,18,8,23,15,17)(9,19,26,34,27,35,28,36)(14,29,30,32,24,33,25,31);(2,4)(5,8,12,6)(7,18,22,17)(9,25,27,30)(10,15,11,16)(13,20,23,21)(14,26,24,28)(19,33,35,29)(31,34,32,36);(5,9)(6,14)(7,19)(8,24)(10,28)(11,26)(12,27)(13,29)(15,25)(16,30)(17,31)(18,32)(20,36)(21,34)(22,35)(23,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371275);