#2940118

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2422 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=gi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2847314);