#511806

order	512
nilpotency class	4
isoclinism	isoclinic class of #511802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2394 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)(7,18)(8,16)(9,15)(10,25)(11,27)(12,17)(13,19)(14,23)(20,28)(21,35)(22,33)(24,32)(26,31)(29,34)(30,36);(5,7)(6,13)(8,19)(9,21)(10,26)(11,20)(12,22)(14,28)(15,30)(16,33)(17,31)(18,25)(23,36)(24,34)(27,29)(32,35);(5,8,12,10)(6,14,17,32)(7,19,22,26)(9,23,11,24)(13,28,31,35)(15,25,27,16)(18,29,33,30)(20,34,21,36);(2,4)(5,9,12,11)(6,15,17,27)(7,20,22,21)(8,23,10,24)(13,29,31,30)(14,25,32,16)(18,28,33,35)(19,34,26,36)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261597, #72280283
maximal normal subgroup of	#72261597, #72280283

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419002);