#461253

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=i,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, 368449);