#272020

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	#29679	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=h,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, 179216);