#394782

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,h^d=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, 301978);