#357280

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2450 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 2450 conjugacy classes of subgroups — too many to draw. Only its 490 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)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22,23,25,24,26,27,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20);(7,12)(9,11)(14,18)(15,19);(6,10)(7,12)(14,18)(17,20)

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264476);