#374037

order	512
nilpotency class	3
isoclinism	isoclinic class of #213778
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2436 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,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, 281233);