#94756

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₄×C₃₂
Frattini, cosocle	C₂²×C₃₂	C₂²
socle	C₂²	#1397
prime radical	C₂³	M₆₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,e^a=ei,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234363, #72286396, #72286831, …
maximal normal subgroup of	#72234363, #72286396, #72286831, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1952);