#466729

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285010, #72285454, #72285571
maximal normal subgroup of	#72285010, #72285454, #72285571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373925);