#549633

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#2825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2680 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=h,e^2=fi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 456829);