#298602

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1263 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1263 conjugacy classes of subgroups — too many to draw. Only its 176 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205798);