#417673

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2479 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255744, #72256257, #72256268, …
maximal normal subgroup of	#72255744, #72256257, #72256268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324869);