#375567

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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:

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

Polycyclic presentation:

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

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, 282763);