#4312013

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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	#15315	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2685 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,22,26,25)(23,28,24,27);(5,6,8,10)(7,9,12,11)(13,15,17,20)(14,16,18,19)(21,23,26,24)(22,27,25,28);(5,7,8,12)(6,9,10,11)(13,14,17,18)(15,19,20,16)(21,22,26,25)(23,27,24,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24,26,23)(22,27,25,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,25,26,22)(23,27,24,28)

Polycyclic presentation:

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

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, 4219209);