#94403

order	512
nilpotency class	5
isoclinism	isoclinic class of #94343
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#367
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

126 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(18,24,25,40)(19,29)(20,34)(21,37)(22,31)(26,45,44,27)(28,48,41,43)(30,38)(32,36)(33,35)(39,46,47,42);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,18,20,26,21,27,19,24,31,41,29,42,30,39,34,48,23,25,36,44,38,45,33,40,22,28,35,46,37,47,32,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225454, #72225503, #72225505
maximal normal subgroup of	#72225454, #72225503, #72225505
maximal quotient of	#72225453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1599);