#6467712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2108 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,22)(17,27)(18,24)(19,23)(21,25)(26,28);(1,2,3,4)(13,15)(14,20)(16,17)(18,25)(19,26)(21,24)(22,27)(23,28);(6,9)(10,12)(13,15,17,26,18,25,19,16)(14,21,27,22,24,20,23,28);(13,16,17,15,18,26,19,25)(14,22,27,20,24,28,23,21);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,24,27)(15,16,25,26)(20,28,21,22)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 6374908);