#1194050

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 1101246);