#2240744

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,c^2=i,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2147940);