#395449

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62853	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3504 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:

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

Polycyclic presentation:

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

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, 302645);