#298938

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33175	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=fh,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206134);