#6345996

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
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	#61448	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1555 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)(3,11,13,9)(4,10,5,15)(7,14,16,12)(17,18,22,19,21,23,29,25)(20,24,28,26,27,30,32,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,16)(9,10)(11,15)(18,23)(19,25)(24,30)(26,31);(2,8)(7,16)(9,11)(10,15)(17,19,22,23,21,25,29,18)(20,26,28,30,27,31,32,24);(1,3)(2,9)(4,14)(5,12)(6,13)(7,10)(8,11)(15,16)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32);(1,4,6,5)(2,10,8,15)(3,12,13,14)(7,9,16,11)(20,27)(24,30)(26,31)(28,32)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=chi,d^a=dh,f^a=fh,c^b=ci,f^b=fh,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339760, #72339782, #72339789
maximal normal subgroup of	#72339760, #72339782, #72339789

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253192);