#346473

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	#62824	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2341 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,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	#72268470, #72268494, #72268496
maximal normal subgroup of	#72268470, #72268494, #72268496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253669);