#461251

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=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, 368447);