#115822

order	512
nilpotency class	3
isoclinism	isoclinic class of #100027
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	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

886 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fi,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23018);