#4308127

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2802 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)(6,8)(10,12)(14,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(9,10,13,12)(11,14,16,15)(19,25)(22,29)(26,31)(30,32);(5,6,7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,2)(3,4)(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215323);