#346469

order	512
nilpotency class	3
isoclinism	isoclinic class of #346446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62819	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2308 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,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.

Relationships

maximal subgroup of	#72213821, #72238412, #72238413, …
maximal normal subgroup of	#72213821, #72238412, #72238413, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253665);