#6437620

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

6304 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239296, #72239541, #72240660, …
maximal normal subgroup of	#72239296, #72239541, #72240660, …
maximal quotient of	#72239358, #72251707, #72257443, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344816);