#195119

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1390 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)(5,6,10,14,11,8,9,13)(7,15,17,16,19,20,18,12)(21,22)(23,26)(24,31,32,27)(25,28)(29,33,35,36)(30,34);(1,2,3,4)(5,7,11,19)(6,12,8,16)(9,17,10,18)(13,15,14,20)(21,23,25,30)(22,26,28,34)(24,29,32,35)(27,33,31,36);(5,8)(6,11)(7,16)(9,14)(10,13)(12,19)(15,18)(17,20)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,35)(34,36);(5,9,11,10)(6,13,8,14)(7,17,19,18)(12,15,16,20)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,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, 102315);