#238870

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60900	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265388, #72265468, #72265472
maximal normal subgroup of	#72265388, #72265468, #72265472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146066);