#473663

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=fgh,c^2=f,d^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380859);