#1572706

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

3227 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)(10,11)(14,15)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28);(6,8)(9,10,13,11)(12,15,16,14)(19,24)(20,23)(27,32)(28,31);(5,6)(7,8)(10,11)(14,15)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27);(9,12)(10,14)(11,15)(13,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1479902);