#577262

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2404 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)(13,14,17,18)(15,16,20,19)(22,25)(23,24);(1,2,4,3)(5,7,9,11)(6,10,12,8)(13,14)(15,16)(17,18)(19,20)(21,22,26,25)(23,27,24,28);(1,3,4,2)(7,11)(8,10)(14,18)(16,19)(21,23,26,24)(22,27,25,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,23)(22,27,25,28);(5,8,9,10)(6,11,12,7)(13,16,17,19)(14,15,18,20)(21,25,26,22)(23,27,24,28)

Polycyclic presentation:

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

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, 484458);