#147674

order	512
nilpotency class	4
isoclinism	isoclinic class of #136643
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1058 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)(5,6,10,16,11,17,9,15)(7,13,18,29,21,30,20,31)(8,22,23,27,25,12,24,14)(19,33,32,36,34,26,35,28);(6,12)(7,18)(8,23)(9,10)(13,28)(14,15)(16,27)(17,22)(19,34)(20,21)(24,25)(26,29)(30,36)(31,33);(2,4)(5,7)(6,13)(8,19)(9,20)(10,18)(11,21)(12,26)(14,28)(15,31)(16,29)(17,30)(22,33)(23,32)(24,35)(25,34)(27,36)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,f^b=fi,g^b=gi,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250558, #72250568, #72250572, …
maximal normal subgroup of	#72250558, #72250568, #72250572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54870);