#6459795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3161 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,15)(16,26,27,28)(17,21,23,24)(18,25,20,22);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,25,19,14,20,22)(16,23,24,28,27,17,21,26);(6,10)(7,11)(13,16,18,24,19,27,20,21)(14,17,22,26,15,23,25,28);(7,11)(9,12)(13,17)(14,21)(15,24)(16,22)(18,26)(19,23)(20,28)(25,27);(16,27)(17,23)(21,24)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366991);