#195253

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19)(21,22,25,27,26,30,24,29)(23,31,33,28,34,35,32,36);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20)(22,27)(24,25)(28,31)(29,30)(32,33)(35,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,34)(22,28,30,36)(24,33,25,32)(27,31,29,35);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 102449);