#268181

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
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	#46225	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1163 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1163 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

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, 175377);