#1095268

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

2256 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)(5,6,10,12)(7,9,8,11)(13,14,17,20)(15,16,19,18)(22,25)(23,27);(5,7,10,8)(6,11,12,9)(13,15,17,19)(14,18,20,16);(6,12)(7,8)(13,16,17,18)(14,19,20,15)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,10,7)(6,11,12,9)(13,16)(14,15)(17,18)(19,20)(21,23,24,27)(22,26,25,28);(5,9,10,11)(6,7,12,8)(13,15)(14,16)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,d^2=g,e^2=h,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, 1002464);