#219791

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,d^c=di,g^c=gh

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, 126987);