#5203684

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=fi,d^2=h,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5110880);