#249078

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36957	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156274);