#4932459

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2158 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)(5,6,11,12)(7,10,9,8)(15,19)(18,20)(23,26)(24,28);(5,7,11,9)(6,8,12,10)(14,17)(18,20)(21,22,25,27)(23,24,26,28);(5,8,11,10)(6,7,12,9)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,26)(27,28);(1,2)(3,4)(5,9,11,7)(6,8,12,10)(21,24)(22,26)(23,27)(25,28);(5,10,11,8)(6,7,12,9)(13,15)(14,18)(16,19)(17,20)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4839655);