#705304

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3190 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=g,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, 612500);