#108770

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(5,6)(7,13,15,19)(8,11)(9,12)(10,14,17,20)(16,18)(21,22,25,28,24,27,31,34)(23,26,30,33,29,32,35,36);(5,7)(6,10,12,17)(8,14)(9,15)(11,13,18,19)(16,20)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,35)(34,36);(1,2,3,4)(22,27)(23,29)(28,34)(30,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=h,f^2=i,b^a=bd,c^a=ce,g^a=gh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15966);