#321576

order	512
nilpotency class	3
isoclinism	isoclinic class of #278051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 512

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

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

Polycyclic presentation:

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

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, 228772);