#4308423

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 50331648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

5252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,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, 4215619);