#423415

order	512
nilpotency class	3
isoclinism	isoclinic class of #423415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15314
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1329 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)(3,8)(4,17,19,28)(5,11)(6,12)(7,13)(9,20,27,22)(10,15)(14,29,30,26)(16,25)(18,23,31,32)(21,24);(1,3)(2,8)(4,18,19,31)(5,21)(6,15)(7,16)(9,26,27,29)(10,12)(11,24)(13,25)(14,20,30,22)(17,32,28,23);(1,4,5,22,7,19,6,20)(2,9,11,28,13,27,12,17)(3,14,15,18,16,30,21,31)(8,23,10,26,25,32,24,29);(1,3)(2,10)(4,14)(5,15)(6,21)(7,16)(8,12)(9,26)(11,25)(13,24)(17,23)(18,22)(19,30)(20,31)(27,29)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215912, #72216029, #72216071, …
maximal normal subgroup of	#72215912, #72216029, #72216071, …
maximal quotient of	C₂×#423415, #72252183, #72314025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330611);