#6346182

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
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

1563 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(17,18,21,19,22,24,29,26)(20,23,28,25,27,30,32,31);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,11)(10,16);(1,4,7,6)(2,9,5,10)(3,12,14,15)(8,11,13,16)(17,19,21,24,22,26,29,18)(20,25,28,30,27,31,32,23);(1,5,7,2)(3,13,14,8)(4,10,6,9)(11,12,16,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,4,7,6)(2,10,5,9)(3,12,14,15)(8,16,13,11)(20,27)(23,30)(25,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340127, #72340131, #72340141
maximal normal subgroup of	#72340127, #72340131, #72340141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253378);