#6381458

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

3124 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)(5,6,9,15,10,16,24,14)(7,17,20,26,21,13,19,11)(8,12,23,30,22,28,35,29)(18,31,34,36,32,27,33,25);(5,7,9,20,10,21,24,19)(6,11,15,17,16,26,14,13)(8,18,23,34,22,32,35,33)(12,25,30,31,28,36,29,27);(2,4)(5,8)(6,12)(7,18)(9,23)(10,22)(11,25)(13,27)(14,29)(15,30)(16,28)(17,31)(19,33)(20,34)(21,32)(24,35)(26,36);(5,7)(6,13)(8,18)(9,20)(10,21)(11,15)(12,27)(14,26)(16,17)(19,24)(22,32)(23,34)(25,30)(28,31)(29,36)(33,35);(8,22)(12,28)(18,32)(23,35)(25,36)(27,31)(29,30)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,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	#72343392, #72343646, #72343682
maximal normal subgroup of	#72343392, #72343646, #72343682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288654);