#92857

order	512
nilpotency class	4
isoclinism	isoclinic class of #92855
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₄

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	C₂×#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×#322 ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 225 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4,10,7,12,15,16,5,13,14,8,6,11,3,9)(17,18)(19,27,30,22)(20,24)(21,26)(23,28,31,32)(25,29);(2,8)(3,14)(4,15)(6,7)(9,11)(10,13)(12,16)(18,22)(19,28)(20,29)(23,25)(24,31)(26,27)(30,32)

Polycyclic presentation:

2^9|a^2=d,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, #72210670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53);