#185272

order	512
nilpotency class	3
isoclinism	isoclinic class of #185267
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	#62826	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 92468);