#352334

order	512
nilpotency class	3
isoclinism	isoclinic class of #352320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

644 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=di,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259530);