#763509

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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	#15278	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(2,4)(9,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22);(6,8)(10,11)(14,15)(18,22)(19,24);(5,6)(7,8)(12,16)(14,15)(18,22)(20,23);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23);(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,23)(22,24)

Polycyclic presentation:

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

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, 670705);