#6371537

order	512
nilpotency class	3
isoclinism	isoclinic class of #60470
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316937
maximal normal subgroup of	#72316937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278733);