#4606593

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
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	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3262 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)(6,9)(7,11)(14,18)(15,19)(21,22,25,26)(23,27,28,24);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28);(7,11)(10,12)(14,18)(16,20)(22,26)(24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,27)(24,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,27)(22,23,26,28)

Polycyclic presentation:

2^9|a^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4513789);