#380138

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

1115 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)(6,11)(7,15)(9,21)(12,25)(13,23)(16,27)(17,30)(18,22)(19,32)(26,35)(28,36)(31,33);(5,6,9,14,10,11,21,24)(7,16,15,25,18,12,22,27)(8,13,19,29,20,23,32,34)(17,28,30,35,31,26,33,36);(5,7,10,18)(6,12,11,16)(8,17,20,31)(9,22,21,15)(13,28,23,26)(14,25,24,27)(19,33,32,30)(29,36,34,35);(5,8)(6,13)(7,17)(9,19)(10,20)(11,23)(12,26)(14,29)(15,30)(16,28)(18,31)(21,32)(22,33)(24,34)(25,35)(27,36)

Polycyclic presentation:

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

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, 287334);