#190045

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302819, #72303054
maximal normal subgroup of	#72302819, #72303054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97241);