C₂×#14673 #544388

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

direct product

C₂×#14673

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

11386 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.

normal subgroups only

This group has 11386 conjugacy classes of subgroups — too many to draw. Only its 1346 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) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210046, #72210050, #72219193, …
maximal normal subgroup of	#72210046, #72210050, #72219193, …
maximal quotient of	#72240531, #72259467, #72262960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451584);