#2738368

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2756 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)(5,6,9,10)(7,11,12,8)(14,17)(15,19)(21,22,25,27)(23,28,26,24);(5,7,9,12)(6,8,10,11)(15,19)(18,20)(21,23,25,26)(22,24,27,28);(5,6,9,10)(7,11,12,8)(13,14)(15,18)(16,17)(19,20)(21,24)(22,26)(23,27)(25,28);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,19)(17,20)(21,22)(23,24)(25,27)(26,28);(5,8,9,11)(6,7,10,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2645564);