#6542626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3315 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:

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

Polycyclic presentation:

2^9|a^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,f^b=fi,d^c=di,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449822);