#384895

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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)(5,6,11,14)(7,12,18,27)(8,19,21,20)(9,10,23,25)(13,24,29,22)(15,30,32,31)(16,17,34,36)(26,35,28,33);(5,7)(6,12)(8,15,21,32)(9,16)(10,17)(11,18)(13,28,29,26)(14,27)(19,31,20,30)(22,33,24,35)(23,34)(25,36);(5,8)(6,13)(7,15)(9,20)(10,24)(11,21)(12,26)(14,29)(16,31)(17,35)(18,32)(19,23)(22,25)(27,28)(30,34)(33,36);(5,9,10,14,11,23,25,6)(7,16,17,27,18,34,36,12)(8,20,24,29,21,19,22,13)(15,31,35,28,32,30,33,26)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,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, 292091);