#412044

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60450	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1344 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250059, #72250071, #72250092, …
maximal normal subgroup of	#72250059, #72250071, #72250092, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319240);