#494873

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

846 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:

(1,2,4,3)(5,6)(7,9)(8,19)(10,18)(11,17)(12,15)(13,22)(14,25)(16,32)(20,30)(21,36)(23,33)(24,31)(26,28)(27,34)(29,35);(1,3,4,2)(5,7,9,6,12,18,10,15)(8,16,20,13,23,28,21,29)(11,17,24,14,27,34,25,31)(19,35,36,26,33,22,30,32);(5,8,10,21,12,23,9,20)(6,13,7,16,15,29,18,28)(11,26,25,35,27,32,24,22)(14,30,17,33,31,36,34,19);(5,9,12,10)(6,7,15,18)(8,20,23,21)(11,24,27,25)(13,16,29,28)(14,17,31,34)(19,30,33,36)(22,32,35,26)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250469, #72250476, #72250487, …
maximal normal subgroup of	#72250469, #72250476, #72250487, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402069);