#6370368

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,d^b=di,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284128, #72308773
maximal normal subgroup of	#72284128, #72308773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277564);