#461296

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

890 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)(5,6,11,22,10,13,9,15)(7,16,18,26,19,30,23,12)(8,14,24,28,21,27,20,29)(17,31,32,35,33,36,34,25);(5,7,9,18,10,19,11,23)(6,12,22,30,13,26,15,16)(8,17,20,32,21,33,24,34)(14,25,28,36,27,35,29,31);(6,13)(8,20)(9,11)(14,28)(16,30)(17,32)(18,23)(21,24)(25,31)(27,29)(33,34)(35,36);(2,4)(5,8,11,24,10,21,9,20)(6,14,22,28,13,27,15,29)(7,17,23,34,19,33,18,32)(12,25,30,36,26,35,16,31)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=ei,e^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

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, 368492);