#195254

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1522 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)(5,6,9,12,10,7,8,11)(13,14,18,21,19,15,17,22)(16,24,26,20,27,28,25,23);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15,19,14)(16,23,27,20)(17,22,18,21)(24,26,28,25);(5,6,10,7)(8,12,9,11)(13,16,19,27)(14,20,15,23)(17,25,18,26)(21,28,22,24);(13,17,19,18)(14,21,15,22)(16,25,27,26)(20,28,23,24)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dhi,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102450);