#391675

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298871);