#381647

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

809 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288843);