#705282

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2786 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fhi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 612478);