#425086

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1018 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:

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

Polycyclic presentation:

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

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, 332282);