#7196785

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=gh,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257061, #72257127, #72257158, …
maximal normal subgroup of	#72257061, #72257127, #72257158, …
maximal quotient of	#72265211

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7103981);