#209115

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

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, 116311);