#383981

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

851 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:

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

Polycyclic presentation:

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

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, 291177);