#378900

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,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,f^c=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, 286096);