#499012

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1072 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321762, #72321775, #72321815, …
maximal normal subgroup of	#72321762, #72321775, #72321815, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406208);