#6676630

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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)(5,6)(7,10)(8,12)(9,11)(13,14,18,19)(15,23,25,20)(16,17,22,21)(24,27,28,26);(5,7,10,6)(8,12,9,11)(13,15,18,25)(14,20,19,23)(16,24,22,28)(17,26,21,27);(2,4)(14,19)(15,23)(16,22)(20,25)(24,27)(26,28);(5,8,10,9)(6,11,7,12)(13,16,18,22)(14,21,19,17)(15,24,25,28)(20,27,23,26);(5,9,10,8)(6,11,7,12)(13,17,18,21)(14,22,19,16)(15,24,25,28)(20,26,23,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583826);