#272026

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	#60924	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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)(3,12)(4,9)(5,14)(6,13)(7,10)(8,11)(15,16)(17,18,23,31,22,27,21,26)(19,24,20,30,29,25,28,32);(1,3,8,15)(2,5,11,4)(6,9,7,14)(10,16,13,12)(17,19)(18,24)(20,23)(21,28)(22,29)(25,27)(26,32)(30,31);(1,4,8,5)(2,9,11,14)(3,13,15,10)(6,16,7,12)(17,20,22,28)(18,25,27,24)(19,21,29,23)(26,32,31,30);(1,5,8,4)(2,3,11,15)(6,16,7,12)(9,13,14,10)(17,21,22,23)(18,26,27,31)(19,20,29,28)(24,30,25,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=hi,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, 179222);