#418514

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	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1706 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)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,24)(18,32)(19,28)(21,26)(23,30)(25,29)(27,35)(31,34)(33,36);(6,12)(8,23)(9,10)(13,26)(15,16)(17,28)(18,33)(19,20)(22,29)(24,25)(27,32)(34,35);(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,17)(14,27)(15,26)(16,28)(22,31)(23,33)(24,34)(25,35)(29,36)(30,32);(2,4)(5,8,9,23,11,25,10,24)(6,14,15,29,16,30,12,22)(7,18,19,33,21,35,20,34)(13,27,26,36,28,32,17,31)

Polycyclic presentation:

2^9|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, 325710);