#7193384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×#3345	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(21,22,27,24)(23,30,31,36)(25,29,26,34)(28,35,33,32);(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23,27,31)(22,28,24,33)(25,32,26,35)(29,36,34,30);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,24,27,22)(23,28,31,33)(25,29,26,34)(30,35,36,32);(7,14)(10,11)(15,17)(19,20)(21,25)(22,29)(24,34)(26,27)(28,33)(32,35);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(22,24)(23,31)(29,34)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309029
maximal normal subgroup of	#72309029

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100580);