#545836

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2020 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)(5,6,9,7)(8,11,12,10)(13,14,17,20)(15,16,19,18);(6,7)(10,11)(13,15,17,19)(14,18,20,16)(21,22,24,26)(23,25,27,28);(1,3,4,2)(6,7)(10,11)(15,19)(16,18)(21,22)(23,25)(24,26)(27,28);(5,7,9,6)(8,11,12,10)(13,16,17,18)(14,19,20,15);(5,8,9,12)(6,10,7,11)(13,14)(15,18)(16,19)(17,20)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fi,d^2=f,e^2=fh,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, 453032);