#379508

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
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	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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)(5,6,11,17,12,18,10,16)(7,13,9,27,21,29,20,15)(8,22,26,14,24,33,25,32)(19,31,35,36,34,30,23,28);(5,7,10,20,12,21,11,9)(6,13,16,15,18,29,17,27)(8,23,25,34,24,35,26,19)(14,30,22,36,32,31,33,28);(5,8,10,26,12,24,11,25)(6,14,16,33,18,32,17,22)(7,19,20,35,21,34,9,23)(13,28,15,31,29,36,27,30);(1,3,4,2)(5,9,11,21,12,20,10,7)(6,15,17,13,18,27,16,29)(8,23,25,34,24,35,26,19)(14,31,22,28,32,30,33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 286704);