#6345905

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	#61432	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339773, #72339786, #72339789
maximal normal subgroup of	#72339773, #72339786, #72339789

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253101);