#181876

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6,3,5,7,8)(9,10)(11,19,22,32)(12,16)(13,17)(14,18)(15,21,27,35)(20,31,33,39)(23,28)(24,29)(25,30)(26,34,37,40)(36,38);(2,5)(6,8)(9,11)(10,15,18,27)(12,20)(13,21)(14,22)(16,26,28,37)(17,19,30,32)(23,33)(24,34)(25,35)(29,31,38,39)(36,40);(9,12,13,24,14,23,25,36)(10,16,17,29,18,28,30,38)(11,20,21,34,22,33,35,40)(15,26,19,31,27,37,32,39);(2,5)(6,8)(12,23)(16,28)(20,33)(24,36)(26,37)(29,38)(31,39)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,e^2=h,f^2=i,b^a=be,d^a=di,g^a=gh,f^b=fh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224691, #72226688
maximal normal subgroup of	#72224691, #72226688

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89072);