#371156

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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)(8,13)(9,24)(10,14)(11,15)(12,19)(17,32)(18,30)(20,27)(21,23)(22,29)(25,28)(26,34)(31,33)(35,36);(1,3,4,2)(5,6)(7,16,18,30)(8,21)(9,24)(10,14)(11,15)(12,20,27,19)(13,23)(17,32,33,31)(22,28)(25,29)(26,35,36,34);(5,7)(6,12)(8,17)(9,20)(10,19)(11,18)(13,26)(14,30)(15,27)(16,24)(21,36)(22,35)(23,33)(25,34)(28,31)(29,32);(5,8,10,22,11,23,9,25)(6,13,14,29,15,21,24,28)(7,17,19,35,18,33,20,34)(12,26,30,32,27,36,16,31)

Polycyclic presentation:

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

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, 278352);