#195318

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1218 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)(5,6,9,12,10,14,8,13)(7,15,16,11,18,19,17,20)(21,22,24,29,26,30,25,27)(23,31,33,28,32,35,34,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,15)(13,19)(14,20)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7)(6,11)(8,17)(9,16)(10,18)(12,19)(13,15)(14,20)(21,23)(22,28)(24,34)(25,33)(26,32)(27,35)(29,31)(30,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,15,20,19)(23,32)(28,36)(31,35)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,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,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102514);