#2079179

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2497 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=f,d^2=f,e^2=fg,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, 1986375);