#191308

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#432 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

505 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)(4,7)(5,9)(6,10)(8,13,16,12)(17,18,22,27,21,26,20,25)(19,23,28,31,29,24,30,32);(2,7,10,9)(8,15,16,11)(17,19)(18,23)(20,30)(21,29)(22,28)(24,26)(25,32)(27,31);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(17,19)(18,24)(20,30)(21,29)(22,28)(23,26)(25,31)(27,32);(17,20,21,22)(18,25,26,27)(19,28,29,30)(23,31,24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=efi,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,d^c=di,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98504);