#379512

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 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)(5,6,12,18,10,16,11,17)(7,19,22,15,21,13,9,29)(8,23,27,33,28,14,26,34)(20,35,25,31,36,30,24,32);(5,7,11,9,10,21,12,22)(6,13,17,15,16,19,18,29)(8,24,26,36,28,25,27,20)(14,31,33,35,23,32,34,30);(5,8,12,26,10,28,11,27)(6,14,18,33,16,23,17,34)(7,20,22,24,21,36,9,25)(13,30,29,31,19,35,15,32);(1,3,4,2)(5,9,11,21,10,22,12,7)(6,15,17,19,16,29,18,13)(8,25,27,36,28,24,26,20)(14,32,34,35,23,31,33,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,g^2=i,h^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,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, 286708);