#271997

order	512
nilpotency class	3
isoclinism	isoclinic class of #271963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60928	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 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)(3,11)(4,8)(5,12)(6,9)(7,10)(13,15)(14,16)(17,18,23,30,22,26,21,25)(19,24,27,31,28,32,20,29);(1,3)(2,4)(5,13)(6,8)(7,14)(9,16)(10,15)(11,12)(17,19)(18,24)(20,21)(22,28)(23,27)(25,29)(26,32)(30,31);(1,4)(2,8)(3,12)(5,11)(6,16)(7,15)(9,14)(10,13)(17,20)(18,24)(19,21)(22,27)(23,28)(25,31)(26,32)(29,30);(1,4)(2,3)(5,16)(6,11)(7,15)(8,12)(9,13)(10,14)(17,21,22,23)(18,25,26,30)(19,27,28,20)(24,31,32,29)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179193);