#460436

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367632);