#6437226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30247	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 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)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,31)(32,35)(33,36);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,25,31,26,32,24,33)(22,27,28,30,29,36,34,35);(1,2,3,4)(7,13)(10,16)(14,19)(17,20);(6,11)(10,16)(12,18)(17,20)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,35,36,30);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,32)(27,36)(30,35)(31,33)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250494, #72250615, #72256945
maximal normal subgroup of	#72250494, #72250615, #72256945
maximal quotient of	#72264400

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344422);