#4386400

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2312 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,16,20,18)(21,22)(23,25)(24,26)(27,28);(6,7)(10,11)(13,15,17,20)(14,18,19,16)(21,23,27,26)(22,24,28,25);(1,2,3,4)(8,12)(10,11)(15,20)(16,18)(21,24,27,25)(22,23,28,26);(5,7,9,6)(8,11,12,10)(13,14)(15,16)(17,19)(18,20)(21,25,27,24)(22,23,28,26);(5,8,9,12)(6,10,7,11)(13,16)(14,15)(17,18)(19,20)(21,26,27,23)(22,24,28,25)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=g,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4293596);