#147472

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54668);