#2309865

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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×#2668	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3124 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2217061);