#119607

order	512
nilpotency class	3
isoclinism	isoclinic class of #100171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=di,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26803);