#358553

order	512
nilpotency class	3
isoclinism	isoclinic class of #184767
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^b=hi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265749);