#6345723

order	512
nilpotency class	3
isoclinism	isoclinic class of #62253
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	#60673	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1633 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)(3,12)(4,10)(5,9)(6,11)(7,13)(8,14)(15,16)(17,18,22,26,23,19,21,25)(20,27,29,31,30,24,28,32);(1,3)(2,7)(4,13)(5,15)(6,14)(8,10)(9,12)(11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,3)(2,8)(4,15)(5,13)(6,14)(7,10)(9,16)(11,12)(17,20,22,29,23,30,21,28)(18,24,26,32,19,27,25,31);(1,4,6,5)(2,9,11,10)(3,13,14,15)(7,12,16,8)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32);(18,19)(24,27)(25,26)(31,32)

Polycyclic presentation:

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

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, 6252919);