#363305

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 270501);