#512742

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₂³
commutator, socle	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#525	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,f^2=g,g^2=h,h^2=i,b^a=be,d^a=di

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231859, #72287227, #72320868, …
maximal normal subgroup of	#72231859, #72287227, #72320868, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419938);