#1142503

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2347 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fgi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1049699);