#362634

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1344 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi

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, 269830);