#195707

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1456 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102903);