#1326440

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15124	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2590 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)(10,11)(12,16)(18,22)(19,20)(25,26,28,31)(27,30,32,29);(1,2,5,3)(4,6,8,7)(9,10,13,11)(12,14,16,15)(17,18)(19,24)(20,23)(21,22)(25,26)(27,30)(28,31)(29,32);(9,10,13,11)(12,14,16,15)(17,19,21,20)(18,23,22,24)(25,27)(26,29)(28,32)(30,31);(1,3,5,2)(4,6,8,7)(9,11,13,10)(12,14,16,15)(17,20,21,19)(18,23,22,24)(25,27)(26,30)(28,32)(29,31);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=fgi,d^2=i,e^2=fh,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, 1233636);