#585568

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2420 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)(5,6,9,11)(7,8,12,10)(14,17)(15,19)(21,22)(23,27)(24,26)(25,28);(5,7,9,12)(6,10,11,8)(14,17)(18,20)(21,22,24,26)(23,27,28,25);(6,11)(7,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,28,27);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(5,8,9,10)(6,7,11,12)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,25,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 492764);