#455179

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,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, 362375);