C₂×#13827 #513755

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄³

Aliases

direct product

C₂×#13827

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

11866 conjugacy classes of subgroups; only the 1858 normal subgroups are shown.

normal subgroups only

This group has 11866 conjugacy classes of subgroups — too many to draw. Only its 1858 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);(7,8,10,9)(12,15)(13,17)(19,20)(21,25)(22,23)(24,26);(3,4,5,6)(7,9,10,8)(12,15)(16,18)(20,23)(21,26);(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26);(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210050, C₂×#11709219, #72219194, …
maximal normal subgroup of	#72210050, C₂×#11709219, #72219194, …
maximal quotient of	C₂×#357285, C₄×#13827, #72258449, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420951);