#347317

order	512
nilpotency class	3
isoclinism	isoclinic class of #347226
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	C₂×2₋¹⁺⁴	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2095 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=h,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,g^b=gh,e^c=ei,f^c=fi,e^d=eh,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268185, #72268253
maximal normal subgroup of	#72268185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254513);