#466434

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=efg,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373630);