#110408

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

908 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,17,20,19)(16,18)(21,22)(23,29)(24,27)(25,33)(26,30)(28,32)(31,34)(35,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,25,31,24,30,32,36)(22,26,28,34,27,29,33,35);(1,2,3,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,27)(26,29)(28,33)(34,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efhi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17604);