#369553

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

726 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=efg,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276749);