#343272

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

949 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 949 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250468);