#233725

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37536	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fhi

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, 140921);