#7196851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#59410	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=gh,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei

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, 7104047);