#346572

order	512
nilpotency class	3
isoclinism	isoclinic class of #346550
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	#62902	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268335, #72268336, #72268357, …
maximal normal subgroup of	#72268335, #72268336, #72268357, …
maximal quotient of	#72282084, #72282185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253768);