#185136

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3626 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

normal subgroups only

This group has 3626 conjugacy classes of subgroups — too many to draw. Only its 346 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257709, #72257858
maximal normal subgroup of	#72257709, #72257858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92332);