#368244

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=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, 275440);