#93097

order	512
nilpotency class	4
isoclinism	isoclinic class of #93083
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
prime radical	C₂³	#332
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

144 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=hi,e^2=fgi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293);