#346465

order	512
nilpotency class	3
isoclinism	isoclinic class of #346446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62826	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2205 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253661);