#496013

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
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 ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

545 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)(5,6,11,16)(7,15,18,34)(8,20,22,28)(9,24,25,14)(10,26,19,12)(13,29,31,17)(21,33,23,36)(27,30,35,32);(5,7,9,19,11,18,25,10)(6,12,14,15,16,26,24,34)(8,17,21,35,22,29,23,27)(13,28,30,33,31,20,32,36);(5,8)(6,13)(7,17)(9,21)(10,27)(11,22)(12,28)(14,30)(15,33)(16,31)(18,29)(19,35)(20,26)(23,25)(24,32)(34,36);(1,3,4,2)(7,18)(8,21,22,23)(10,19)(12,26)(13,30,31,32)(15,34)(17,27,29,35)(20,33,28,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,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^b=di,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, 403209);