#185516

order	512
nilpotency class	3
isoclinism	isoclinic class of #185503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3517	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2410 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)(3,11,13,7)(4,14,15,8)(5,9)(6,10)(12,16)(17,18)(19,26,27,23)(20,22)(21,25)(24,29)(28,31,32,30);(1,3,6,13)(2,7,10,11)(4,12,15,5)(8,16,14,9)(18,22)(20,29)(23,30)(24,25)(26,31)(28,32);(1,4)(2,8)(3,12)(5,13)(6,15)(7,16)(9,11)(10,14)(17,19)(18,23)(20,28)(21,27)(22,30)(24,31)(25,26)(29,32);(1,5,6,12)(2,9,10,16)(3,4,13,15)(7,8,11,14)(19,27)(23,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,e^2=i,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, 92712);