#495225

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402421);