#493783

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249886, #72249908, #72249947, …
maximal normal subgroup of	#72249886, #72249908, #72249947, …
maximal quotient of	#72265692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400979);