#465828

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	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 373024);