#3135088

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
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

2797 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)(10,13)(11,15)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,31,30,32);(6,8)(11,15)(14,16)(18,19)(21,29)(23,25)(27,30);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32);(1,2,3,4)(9,11,12,15)(10,14,13,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31);(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 3042284);