#846487

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 753683);