#395556

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
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	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2155 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=fi,h^c=hi,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, 302752);