#346476

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

2278 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:

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

Polycyclic presentation:

2^9|a^2=i,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.

Relationships

maximal subgroup of	#72213843, #72232843, #72234684, …
maximal normal subgroup of	#72213843, #72232843, #72234684, …
maximal quotient of	C₂×#346476, #72261542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253672);