#92858

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

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

normal subgroups only

This group has 175 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)(3,11,14,6)(4,8)(5,10)(7,12,15,16)(9,13)(17,18,20,26,22,27,19,29,23,28,30,24,21,25,31,32);(2,6)(3,12)(4,13)(7,9)(8,15)(10,11)(14,16)(17,19,23,31)(18,24,28,26)(20,21,30,22)(25,32,27,29)

Polycyclic presentation:

2^9|a^2=d,b^2=h,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, #72210671

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54);