#456515

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1129 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)(7,18)(8,22)(9,14,27,33)(10,16)(11,15)(12,17)(13,20)(19,30,23,34)(21,29)(24,36,35,28)(25,32,26,31);(1,2)(3,4)(5,7,10,20,12,8,11,21)(6,13,15,18,17,29,16,22)(9,19,26,24,27,23,25,35)(14,28,32,34,33,36,31,30);(2,4)(5,8)(6,13)(7,12)(9,24)(10,20)(11,21)(14,30)(15,22)(16,18)(17,29)(19,26)(23,25)(27,35)(28,31)(32,36)(33,34);(5,9)(6,14)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28)(15,32)(16,31)(17,33)(18,34)(20,24)(21,35)(22,30)(29,36)

Polycyclic presentation:

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

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, 363711);