#2822362

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×#2407	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3072 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)(13,14,18,20)(15,17,16,19)(22,26)(23,24);(2,4)(5,6,8,10)(7,9,12,11)(13,15,18,16)(14,19,20,17)(21,22,25,26)(23,27,24,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,16)(14,17,20,19)(21,23,25,24)(22,27,26,28);(5,6)(7,9)(8,10)(11,12)(13,16,18,15)(14,17,20,19)(21,24,25,23)(22,27,26,28);(5,7)(6,9)(8,12)(10,11)(13,17)(14,16)(15,20)(18,19)(21,22)(23,27)(24,28)(25,26)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=f,d^2=f,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, 2729558);