#2923258

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(9,10,13,12)(11,15,16,14)(17,18,23,27)(19,25,20,24)(21,32,26,22)(28,29,30,31);(6,8)(10,12)(14,15)(17,19,23,20)(18,24,27,25)(21,30,26,28)(22,31,32,29);(5,6,7,8)(17,20,23,19)(18,25,27,24)(21,30,26,28)(22,31,32,29);(1,2,3,4)(9,11,13,16)(10,14,12,15)(17,21)(18,22)(19,28)(20,30)(23,26)(24,29)(25,31)(27,32);(9,12,13,10)(11,14,16,15)(17,22)(18,26)(19,29)(20,31)(21,27)(23,32)(24,30)(25,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=gi,e^2=fg,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, 2830454);