#414032

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15160
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1219 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)(3,13)(4,9)(5,20)(6,11)(7,12)(8,15)(10,21)(14,29)(16,24)(17,23)(18,27)(19,28)(22,26)(25,30)(31,32);(1,3,6,15,7,16,21,5)(2,8,10,13,12,20,11,24)(4,17,18,30,19,31,22,14)(9,25,26,23,28,29,27,32);(1,4)(2,9)(3,14)(5,17)(6,22)(7,19)(8,23)(10,27)(11,26)(12,28)(13,25)(15,31)(16,30)(18,21)(20,32)(24,29);(1,5,21,16,7,15,6,3)(2,8,10,13,12,20,11,24)(4,17,18,30,19,31,22,14)(9,23,27,25,28,32,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318876, #72318883, #72318981, …
maximal normal subgroup of	#72318876, #72318883, #72318981, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321228);