#347218

order	512
nilpotency class	3
isoclinism	isoclinic class of #346934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63016	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271563, #72272121
maximal normal subgroup of	#72271563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254414);