#195123

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#3100	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102319);