#2044080

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2157 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1951276);