#147592

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,11,17,12,18,10,16)(7,19,22,47,23,48,31,60)(8,14,26,41,27,42,32,44)(9,28,29,45,30,15,33,43)(13,34,37,21,38,52,36,53)(20,46,50,66,51,67,24,54)(25,56,58,64,59,40,57,65)(35,61,39,49,62,55,63,68);(5,7,12,23)(6,13,18,38)(8,24,27,50)(9,21,30,53)(10,31,11,22)(14,39,42,63)(15,19,28,48)(16,37,17,36)(20,32,51,26)(25,55,59,61)(29,52,33,34)(35,44,62,41)(40,54,56,66)(43,60,45,47)(46,65,67,64)(49,58,68,57);(2,4)(5,8,12,27)(6,14,18,42)(7,20,23,51)(9,25,30,59)(10,32,11,26)(13,35,38,62)(15,40,28,56)(16,44,17,41)(19,46,48,67)(21,49,53,68)(22,50,31,24)(29,58,33,57)(34,61,52,55)(36,63,37,39)(43,65,45,64)(47,66,60,54)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54788);