#6537109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307763, #72307872, #72308071, …
maximal normal subgroup of	#72307763, #72307872, #72308071, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444305);