#92859

order	512
nilpotency class	4
isoclinism	isoclinic class of #92855
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1270
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	C₂×#322	C₂²
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×#322 ▷ C₂×C₄² ▷ C₂² ▷ C₁

183 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=hi,c^2=fhi,d^2=g,e^2=i,f^2=hi,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fhi,g^b=gh,d^c=dhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72210669, #72210672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55);