#379481

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 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)(7,12)(8,19)(9,24)(10,14)(11,15)(13,22)(16,33)(17,28)(18,29)(20,26)(21,32)(23,31)(25,27)(30,34)(35,36);(5,7,10,17,11,18,9,25)(6,12,14,28,15,29,24,27)(8,20,23,34,21,35,22,16)(13,30,19,36,31,33,32,26);(5,8,10,22,11,21,9,23)(6,13,14,32,15,31,24,19)(7,16,17,35,18,34,25,20)(12,26,28,33,29,36,27,30);(1,3,4,2)(5,6)(7,12)(8,13)(9,24)(10,14)(11,15)(16,26)(17,28)(18,29)(19,23)(20,30)(21,31)(22,32)(25,27)(33,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286677);