#6345651

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1541 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252847);