#500676

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33517	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1243 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265538, #72265562, #72265636
maximal normal subgroup of	#72265538, #72265562, #72265636

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407872);