#6380341

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287537);