#439067

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286957, #72287334, #72287360
maximal normal subgroup of	#72286957, #72287334, #72287360

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346263);