#364105

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,g^c=gi,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, 271301);