#191304

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
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 ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

508 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dhi,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, 98500);