#101427

order	512
nilpotency class	3
isoclinism	isoclinic class of #101351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#23930	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

838 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257795, #72257821, #72257822
maximal normal subgroup of	#72257795, #72257821, #72257822
maximal quotient of	#72265092, #72265096, #72265549, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8623);