#1007460

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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

3214 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)(7,12)(8,9)(10,11)(14,18)(19,20)(23,27)(26,28);(6,11)(7,8)(13,14)(15,20)(16,19)(17,18)(22,25)(26,28);(5,7,10,8)(6,9,11,12)(13,15,17,16)(14,19,18,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,16,17,15)(14,19,18,20)(21,23)(22,26)(24,27)(25,28);(5,9)(6,7)(8,11)(10,12)(13,14)(15,20)(16,19)(17,18)

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=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, 914656);