#1082700

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

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

3594 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)(5,6)(7,8)(9,10)(11,12)(14,15)(16,20)(21,22,25,27)(23,24,28,26);(1,2)(3,4)(6,10)(7,12)(13,14,17,15)(16,18,20,19)(21,23,25,28)(22,26,27,24);(1,2,3,4)(6,10)(8,11)(16,20)(18,19)(22,27)(24,26);(5,7,9,12)(6,11,10,8)(13,15,17,14)(16,19,20,18)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,8,9,11)(6,12,10,7)(13,16)(14,18)(15,19)(17,20)(21,24)(22,23)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=h,d^2=i,e^2=fghi,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, 989896);