#388278

order	512
nilpotency class	3
isoclinism	isoclinic class of #30496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

1061 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=i,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,f^c=fi,g^c=gi,e^d=ei,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, 295474);