#395178

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1740 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=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, 302374);