#418516

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1700 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)(7,15)(8,22)(9,26)(10,17)(11,16)(12,18)(13,20)(14,24)(19,30)(21,29)(23,36)(25,33)(27,32)(28,35)(31,34);(5,7,12,21)(6,13,18,26)(8,19,25,35)(9,11,20,10)(14,30,32,28)(15,16,29,17)(22,31,33,36)(23,27,34,24);(5,8)(6,14)(7,19)(9,23)(10,24)(11,27)(12,25)(13,28)(15,31)(16,22)(17,33)(18,32)(20,34)(21,35)(26,30)(29,36);(2,4)(5,9,10,7,12,20,11,21)(6,15,16,26,18,29,17,13)(8,23,24,19,25,34,27,35)(14,31,22,30,32,36,33,28)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325712);