#465830

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=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, 373026);