#4420583

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2635	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2606 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)(5,6)(7,12)(8,10)(9,11)(14,18)(15,16)(21,22,25,28)(23,24,27,26);(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,19)(17,18)(21,23)(22,26)(24,28)(25,27);(13,15,17,16)(14,19,18,20)(21,24,25,26)(22,27,28,23);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16,17,15)(14,19,18,20)(21,23)(22,24)(25,27)(26,28);(5,8,9,7)(6,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=h,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4327779);