#191311

order	512
nilpotency class	3
isoclinism	isoclinic class of #8803
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#383
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98507);