#392215

order	512
nilpotency class	3
isoclinism	isoclinic class of #184773
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^c=hi

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, 299411);