#392156

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299352);