#496127

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

757 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)(6,11,14,24)(8,19)(9,21)(12,22,26,34)(13,23,15,29)(17,32)(18,31)(20,27)(25,33,28,35)(30,36);(5,6,8,11,9,15,19,23,10,14,20,24,21,13,27,29)(7,12,17,22,18,25,32,35,16,26,30,34,31,28,36,33);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,22)(13,28)(14,26)(15,25)(19,32)(20,30)(21,31)(23,35)(24,34)(27,36)(29,33);(6,13,14,15)(7,16)(8,20)(11,23,24,29)(12,25,26,28)(18,31)(19,27)(22,33,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224826, #72250867, #72265409, …
maximal normal subgroup of	#72224826, #72250867, #72265409, …
maximal quotient of	#72224823, #72287318, #72309431, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403323);