#391579

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

651 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)(7,18)(8,24)(9,15)(10,32)(11,16)(12,17)(13,21)(14,27)(19,31)(20,28)(22,29)(23,36)(25,33)(26,34)(30,35);(5,7,12,23)(6,13,17,29)(8,25,27,35)(9,24,31,14)(10,22,11,21)(15,28,19,26)(16,36,32,18)(20,30,34,33);(5,8,10,26,12,27,11,28)(6,14,16,34,17,24,32,20)(7,19,21,35,23,15,22,25)(9,29,30,36,31,13,33,18);(5,9,11,30,12,31,10,33)(6,15,16,35,17,19,32,25)(7,20,22,14,23,34,21,24)(8,18,28,29,27,36,26,13)

Polycyclic presentation:

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

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, 298775);