#402431

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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 ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1340 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)(5,6,9,12)(7,11,8,10)(13,14,16,18)(15,17,19,20)(21,22,25,26)(23,30,32,36)(24,28,33,29)(27,35,34,31);(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,19)(18,20)(22,26)(27,34)(28,29)(30,36);(5,7,9,8)(6,11,12,10)(15,19)(17,20)(21,23,24,31,25,32,33,35)(22,27,29,30,26,34,28,36);(2,4)(5,8,9,7)(6,10,12,11)(21,24,25,33)(22,28,26,29)(23,31,32,35)(27,36,34,30)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309627);