#373558

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1080 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280754);