#1452031

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2073 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=f,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1359227);