#385204

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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, 292400);