#272103

order	512
nilpotency class	3
isoclinism	isoclinic class of #272091
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	#3460	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,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, 179299);