#119952

order	512
nilpotency class	3
isoclinism	isoclinic class of #99835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,g^b=gh,e^c=ei,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27148);