#6345015

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60570	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^d=eh

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321275, #72343921
maximal normal subgroup of	#72321275, #72343921
maximal quotient of	#72343940

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252211);