#185809

order	512
nilpotency class	3
isoclinism	isoclinic class of #185503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2549 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93005);