#1294823

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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

3140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,3)(6,7)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,27,28,24)(20,30,31,25)(21,29,32,26);(9,10,12,14)(11,13,16,15)(18,23)(19,28)(25,30)(26,29);(1,2,4,3)(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30);(5,6,8,7)(17,20,22,31)(18,25,23,30)(19,29,28,26)(21,24,32,27);(1,3,4,2)(5,7,8,6)(9,11,12,16)(10,13,14,15)(17,21,22,32)(18,26,23,29)(19,25,28,30)(20,27,31,24)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fgh,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1202019);