#2046708

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

2185 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fghi,c^2=fg,d^2=fgh,e^2=fgh,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, 1953904);