#119606

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,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, 26802);