#2044104

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1898 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,24)(19,21,20,23)(25,26,29,28)(27,30,31,32);(1,3,6,5)(2,7,4,8)(10,13)(11,15)(17,19,22,20)(18,23,24,21)(27,31)(30,32);(1,2,6,4)(3,8,5,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,19)(18,23,24,21)(25,27,29,31)(26,30,28,32);(1,4,6,2)(3,7,5,8)(9,12,14,16)(10,15,13,11)(25,28,29,26)(27,32,31,30);(1,5,6,3)(2,7,4,8)(9,13,14,10)(11,16,15,12)(17,21,22,23)(18,19,24,20)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=fh,d^2=h,e^2=fg,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, 1951300);