#494462

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1460 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)(5,6,10,16,25,32,23,33,11,13,9,15,8,14,24,34)(7,17,21,31,18,36,26,12,20,30,19,29,22,35,27,28);(1,2,3,4)(5,7,11,20)(6,12,13,28)(8,18,25,22)(9,26,10,27)(14,29,32,31)(15,35,16,36)(17,34,30,33)(19,24,21,23);(6,13)(9,10)(12,28)(14,32)(19,21)(23,24)(26,27)(29,31);(5,8,11,25)(6,14,13,32)(7,18,20,22)(9,23,10,24)(12,29,28,31)(15,33,16,34)(17,36,30,35)(19,27,21,26)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=gi

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, 401658);