#411284

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

948 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fgi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318480);