#146493

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1820
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=ci,f^b=fh,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240802, #72257229, #72280249
maximal normal subgroup of	#72240802, #72257229, #72280249
maximal quotient of	#72240805

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53689);