#6345403

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	#60561	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1701 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,25)(22,28)(26,31)(29,32);(2,9)(7,16)(8,13)(10,15)(18,23)(22,28)(24,30)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339789, #72339810, #72339814
maximal normal subgroup of	#72339789, #72339810, #72339814

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252599);