#209605

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
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	#18128	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116801);