#147423

order	512
nilpotency class	4
isoclinism	isoclinic class of #13215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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)(7,16)(8,21)(9,25)(10,15)(11,18)(12,23)(13,30)(14,22)(17,34)(19,28)(20,35)(24,31)(26,33)(27,36)(29,32);(5,7,8,18,22,26,30,32,10,19,24,36,9,20,23,34)(6,11,13,28,25,17,21,33,15,27,12,16,14,29,31,35);(2,4)(5,6)(7,17,19,29)(8,12)(9,14)(10,15)(11,26,27,20)(13,24)(16,32,28,34)(18,35,36,33)(21,23)(22,25)(30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249985, #72249995, #72250008, …
maximal normal subgroup of	#72249985, #72249995, #72250008, …
maximal quotient of	#72265654, #72280326

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54619);