#115938

order	512
nilpotency class	3
isoclinism	isoclinic class of #99923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 23134);