#376262

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2408
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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)(5,6,11,16)(7,17,19,30)(8,12,21,13)(9,22,26,35)(10,27,24,15)(14,31,32,25)(18,29,34,28)(20,33,23,36);(5,7,10,8,11,19,24,21)(6,12,15,17,16,13,27,30)(9,18,25,20,26,34,31,23)(14,28,22,33,32,29,35,36);(5,8,10,19,11,21,24,7)(6,13,15,30,16,12,27,17)(9,23,25,18,26,20,31,34)(14,28,22,33,32,29,35,36);(1,3,4,2)(5,9,24,31,11,26,10,25)(6,14,27,35,16,32,15,22)(7,18,21,23,19,34,8,20)(12,28,30,36,13,29,17,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=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, 283458);