#612419

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#17068	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519615);