#545165

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2556 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)(6,8)(10,11)(14,16)(18,19)(21,22,27,25)(23,24,28,26);(1,3,4,2)(5,6,9,8)(7,10,12,11)(13,14,17,16)(15,19,20,18)(21,23)(22,26)(24,25)(27,28);(5,7,9,12)(6,10,8,11)(13,14,17,16)(15,19,20,18)(21,24,27,26)(22,28,25,23);(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,18,16,19)(21,25,27,22)(23,24,28,26);(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,19,20,18)(21,26,27,24)(22,28,25,23)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=gh,e^2=f,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, 452361);