#406358

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1333 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,19,24,18,23)(15,25,28,37,29,36,27,20)(16,30,32,21,34,39,33,40)(26,35,41,38,42,43,31,44);(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,31)(17,27)(18,28)(19,29)(21,38)(22,36)(23,25)(24,37)(26,33)(30,43)(32,42)(34,41)(35,39)(40,44);(1,2,3,4)(13,16,17,32,19,34,18,33)(14,21,23,30,24,40,22,39)(15,26,28,41,29,42,27,31)(20,35,36,44,37,43,25,38);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,34,33)(20,36,37,25)(21,39,40,30)(26,31,42,41)(35,44,43,38)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 313554);