#2502763

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2118 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=fg,d^2=fg,e^2=fghi,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, 2409959);