#382698

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×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	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 289894);