#2329533

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2146 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)(7,9)(8,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22,25,24)(23,26,28,27);(6,7)(10,11)(14,18)(15,20)(21,22,25,24)(23,27,28,26);(8,12)(10,11)(13,15)(14,19)(16,18)(17,20)(21,23,25,28)(22,26,24,27);(1,2,3,4)(5,7,9,6)(8,11,12,10)(21,22,25,24)(23,26,28,27);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,20,18,15)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2236729);