#1293348

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3280 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=f,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, 1200544);