#496012

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=egh,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=dh,f^c=fh,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, 403208);