#6381460

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60366	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3122 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)(5,6,9,16,10,14,25,15)(7,13,20,27,19,33,18,11)(8,21,24,31,23,12,28,32)(17,34,36,26,22,30,35,29);(5,7,9,20,10,19,25,18)(6,11,16,13,14,27,15,33)(8,22,24,35,23,17,28,36)(12,29,32,34,21,26,31,30);(2,4)(5,8)(6,12)(7,17)(9,24)(10,23)(11,26)(13,30)(14,21)(15,31)(16,32)(18,35)(19,22)(20,36)(25,28)(27,29)(33,34);(5,7)(6,13)(8,17)(9,20)(10,19)(11,15)(12,30)(14,33)(16,27)(18,25)(21,34)(22,23)(24,36)(26,31)(28,35)(29,32);(6,14)(8,23)(11,27)(13,33)(15,16)(17,22)(24,28)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ci,d^b=df,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343474, #72343653, #72343682
maximal normal subgroup of	#72343474, #72343653, #72343682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288656);