#3469107

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2872 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)(6,9)(7,11)(13,14)(15,20)(16,17)(18,19)(23,27)(25,28);(6,9)(10,12)(14,19)(15,16)(21,22,24,26)(23,25,27,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,20,19,17)(21,23)(22,25)(24,27)(26,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,19,20)(21,22)(23,25)(24,26)(27,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,25)(24,27)(26,28)

Polycyclic presentation:

2^9|b^2=fgh,c^2=f,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3376303);