#371638

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

946 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,23,25,28,35,30)(24,31,34,32,33,27,36,29);(5,7,9,8)(6,10,12,11)(14,17)(15,20)(21,23,26,28,25,30,35,22)(24,32,34,27,33,29,36,31);(2,4)(5,7,9,8)(6,11,12,10)(13,15,16,20)(14,18,17,19)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36);(5,8,9,7)(6,11,12,10)(14,17)(18,19)(24,33)(27,31)(29,32)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=e,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di

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, 278834);