#461302

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

853 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:

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

Polycyclic presentation:

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

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, 368498);