#195276

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1422 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)(7,10)(8,15,16,11)(9,12)(13,17,19,20)(14,18)(21,22,25,28,26,30,24,29)(23,31,34,27,32,36,33,35);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,20)(16,19)(21,23)(22,27)(24,34)(25,33)(26,32)(28,31)(29,36)(30,35);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20)(22,28)(24,25)(27,31)(29,30)(33,34)(35,36);(6,12)(10,18)(11,15)(17,20)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

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

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, 102472);