#6345443

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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

1636 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287392, #72318783, #72319880, …
maximal normal subgroup of	#72287392, #72318783, #72319880, …
maximal quotient of	#72343924, #72343928

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252639);