#346481

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	#62820	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2346 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,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, #72232836, #72232843, …
maximal normal subgroup of	#72213843, #72232836, #72232843, …
maximal quotient of	C₂×#346481, #72261542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253677);