#417712

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15257
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215981, #72216141, #72219688, …
maximal normal subgroup of	#72215981, #72216141, #72219688, …
maximal quotient of	C₂×#417712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324908);