#805021

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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

2397 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 712217);