#989425

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3168 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 896621);