#4423122

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=h,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, 4330318);