#6491015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3342 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)(7,11)(10,12)(13,14)(15,24)(16,21)(17,22)(18,23)(19,26)(20,25)(27,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,19,23,20)(16,25,17,26)(21,24,22,28);(6,9)(10,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,28)(22,24);(6,9)(10,12)(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,24,20,28);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,22,23,21)(15,25,27,26)(19,28,20,24)

Polycyclic presentation:

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

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, 6398211);