#493440

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3280 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)(5,6,10,16,11,17,7,14)(8,13,20,27,22,29,18,30)(9,15,19,26,24,31,23,12)(21,28,32,36,34,35,33,25);(5,7,11,10)(6,12,17,26)(8,18,22,20)(9,23,24,19)(13,25,29,36)(14,15,16,31)(21,33,34,32)(27,35,30,28);(2,4)(5,8)(6,13)(7,18)(9,21)(10,20)(11,22)(12,25)(14,30)(15,28)(16,27)(17,29)(19,32)(23,33)(24,34)(26,36)(31,35);(5,7,11,10)(6,14,17,16)(8,20,22,18)(9,19,24,23)(12,15,26,31)(13,27,29,30)(21,33,34,32)(25,35,36,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302925, #72303035
maximal normal subgroup of	#72302925, #72303035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400636);