#6345157

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	#60673	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1640 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18,22,25,21,24,30,32)(19,26,28,29,27,23,31,20);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,23)(20,25)(21,27)(22,28)(24,26)(29,32)(30,31);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(17,20,22,26,21,29,30,23)(18,19,25,28,24,27,32,31);(19,27)(20,29)(23,26)(28,31);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321617, #72321889, #72321907
maximal normal subgroup of	#72321617, #72321889, #72321907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252353);