C₂×#15155 #608401

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

direct product

C₂×#15155

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

7166 conjugacy classes of subgroups; only the 1546 normal subgroups are shown.

normal subgroups only

This group has 7166 conjugacy classes of subgroups — too many to draw. Only its 1546 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) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268798
Frattini subgroup of	#72268792, #72268798, #72268817
Fitting subgroup of	C₂×C₄⋊(#399⋊C₃)
maximal subgroup of	#72266417, #72266422, #72266423, …
maximal normal subgroup of	#72266417, #72266422, #72266423, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515597);