#6345798

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61427	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1651 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)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(20,28)(24,26)(29,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234372, #72321210, #72321227, …
maximal normal subgroup of	#72234372, #72321210, #72321227, …
maximal quotient of	#72320150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252994);