#4987001

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2400 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)(5,6,10,11)(7,8,9,12)(14,17)(15,19)(23,28)(24,26);(5,7,10,9)(6,8,11,12)(13,14,16,17)(15,18,19,20)(21,22,25,27)(23,26,28,24);(6,11)(8,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,28)(22,24,27,26);(1,2,3,4)(5,8,10,12)(6,7,11,9)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(5,9,10,7)(6,12,11,8)(13,14)(15,18)(16,17)(19,20)(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=h,d^2=fhi,e^2=fg,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, 4894197);