#696673

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2250 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)(4,8)(5,6)(10,13)(14,15)(17,18)(19,20)(21,23)(22,24)(25,26,30,29)(27,32,31,28);(2,6)(4,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,24)(18,22,23,20)(25,26,30,29)(27,28,31,32);(2,6)(3,8)(10,13)(14,15)(19,24)(20,22)(25,27)(26,28)(29,32)(30,31);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28,30,32)(26,31,29,27);(1,4,5,7)(2,8,6,3)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,23)(22,24)(25,29,30,26)(27,32,31,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 603869);