#644678

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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
prime radical	C₂×#2856	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3270 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)(5,6,8,9)(7,11,12,10)(14,16)(18,20)(21,22,25,27)(23,28,24,26);(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20)(21,23,25,24)(22,26,27,28);(1,2,3,4)(6,9)(10,11)(15,19)(18,20)(22,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20)(21,22)(23,26)(24,28)(25,27);(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,20,19,18)(21,24,25,23)(22,28,27,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

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, 551874);