#6345882

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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 ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1601 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)(3,9,12,15)(4,8,5,14)(7,13,16,11)(17,18,22,26,23,19,21,25)(20,24,30,32,28,27,29,31);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,28)(29,32)(30,31);(1,4,6,5)(2,8,10,14)(3,11,12,13)(7,15,16,9)(17,20,22,30,23,28,21,29)(18,24,26,32,19,27,25,31);(1,3)(2,9)(4,13)(5,11)(6,12)(7,14)(8,16)(10,15)(20,28)(24,27)(29,30)(31,32);(1,4,6,5)(2,8,10,14)(3,11,12,13)(7,15,16,9)(17,21,23,22)(18,25,19,26)(20,29,28,30)(24,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322235, #72338327, #72338418, …
maximal normal subgroup of	#72322235, #72338327, #72338418, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253078);