#1291434

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2221 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)(5,6,9,8)(7,11,12,10)(22,25)(23,27);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(22,25)(26,28);(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,9,6)(7,11,12,10);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=f,e^2=fh,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, 1198630);