#417803

order	512
nilpotency class	3
isoclinism	isoclinic class of #417802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2679 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)(8,20)(9,12)(10,27)(11,15)(13,18)(14,24)(17,29)(19,30)(21,31)(22,23)(25,32)(26,28)(33,36)(34,35);(1,2,3,4)(5,7,10,18,12,19,11,28)(6,13,15,16,9,26,27,30)(8,21,23,17,25,33,24,34)(14,31,32,35,22,36,20,29);(5,8)(6,14)(7,17)(9,22)(10,23)(11,24)(12,25)(13,29)(15,32)(16,31)(18,33)(19,34)(20,27)(21,28)(26,35)(30,36);(2,4)(5,9,12,6)(7,16,19,30)(8,22,25,14)(10,15,11,27)(13,28,26,18)(17,31,34,36)(20,23,32,24)(21,35,33,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341337, #72341344, #72341354
maximal normal subgroup of	#72341337, #72341344, #72341354

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324999);