#7196790

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

1855 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)(21,22,26,35,25,30,24,29)(23,28,32,27,34,31,33,36);(7,13)(8,16)(10,17)(11,15)(21,23)(22,27)(24,32)(25,34)(26,33)(28,35)(29,31)(30,36);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23)(22,28)(24,33)(25,34)(26,32)(27,35)(29,36)(30,31);(7,13)(10,17)(14,19)(18,20)(21,22,24,29,25,30,26,35)(23,31,33,27,34,28,32,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,25,26)(22,29,30,35)(23,32,34,33)(27,31,36,28)

Polycyclic presentation:

2^9|a^2=fhi,c^2=gh,d^2=f,e^2=hi,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	#72257048, #72257134, #72257158
maximal normal subgroup of	#72257048, #72257134, #72257158
maximal quotient of	#72264526

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7103986);