#1360324

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

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, 1267520);