#493779

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320324, #72320360, #72320546, …
maximal normal subgroup of	#72320324, #72320360, #72320546, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400975);