#6668299

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2790 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)(7,11)(8,10)(9,12)(13,14,19,18)(15,20,24,28)(16,21,27,26)(17,25,23,22);(5,7,8,12)(6,9,10,11)(13,15,19,24)(14,20,18,28)(16,25,27,22)(17,21,23,26);(2,4)(13,16,19,27)(14,21,18,26)(15,23,24,17)(20,22,28,25);(6,10)(9,11)(13,14,19,18)(15,20,24,28)(16,26,27,21)(17,22,23,25);(7,12)(9,11)(13,17,19,23)(14,22,18,25)(15,16,24,27)(20,26,28,21)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575495);