#385203

order	512
nilpotency class	3
isoclinism	isoclinic class of #30871
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,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, 292399);