#498887

order	512
nilpotency class	4
isoclinism	isoclinic class of #498886
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3026	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2307 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,29)(18,30)(19,26)(20,25)(21,28)(22,27)(23,31)(24,33)(32,36)(34,35);(5,7)(6,11)(8,16)(9,17)(10,18)(12,23)(13,24)(14,15)(19,31)(20,32)(21,34)(22,33)(25,35)(26,29)(27,30)(28,36);(2,4)(5,8,10,20)(6,12,13,25)(7,16,18,32)(9,21,22,19)(11,23,24,35)(14,28,27,26)(15,29,30,36)(17,31,33,34);(6,13)(11,24)(12,25)(14,27)(15,30)(23,35)(26,28)(29,36)

Polycyclic presentation:

2^9|c^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,h^b=hi,e^c=eh,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279131, #72279160, #72279215, …
maximal normal subgroup of	#72279131, #72279160, #72279215, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406083);