#384896

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=efi,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, 292092);