#2972697

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2684 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=fi,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, 2879893);