#246951

order	512
nilpotency class	3
isoclinism	isoclinic class of #246927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18152	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1163 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(19,25)(21,29)(23,31)(24,28)(26,32)(27,30);(1,3,6,13)(2,7,10,15)(4,14,5,12)(8,16,9,11)(17,18,22,20)(19,23,27,26)(21,28,29,24)(25,32,30,31);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(17,19)(18,23)(20,26)(21,30)(22,27)(24,32)(25,29)(28,31);(2,9)(4,5)(7,11)(8,10)(12,14)(15,16)(17,20,22,18)(19,26,27,23)(21,24,29,28)(25,31,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305968, #72305970, #72305977
maximal normal subgroup of	#72305968, #72305970, #72305977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154147);