#209113

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	#3459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 116309);