#4861912

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 2097152
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4173 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:

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

Polycyclic presentation:

2^9|b^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4769108);