#390996

order	512
nilpotency class	3
isoclinism	isoclinic class of #32700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284974, #72285164
maximal normal subgroup of	#72284974, #72285164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298192);