#390340

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

747 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)(5,6,11,14)(7,15,19,25)(8,20,24,13)(9,10,26,21)(12,27,17,18)(16,33,35,32)(22,23,31,28)(29,36,30,34);(5,7)(6,12)(8,16,24,35)(9,25)(10,27)(11,19)(13,30,20,29)(14,17)(15,26)(18,21)(22,32,31,33)(23,36,28,34);(1,3,4,2)(5,8,21,23,11,24,10,28)(6,13,26,31,14,20,9,22)(7,16,27,34,19,35,18,36)(12,29,25,33,17,30,15,32);(5,9,10,14,11,26,21,6)(7,17,18,25,19,12,27,15)(8,22,28,20,24,31,23,13)(16,30,36,33,35,29,34,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=efh,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 297536);