C₂×#12172 #484650

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12172

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

904 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486333, C₂×#485590, C₂×#486084, …
maximal normal subgroup of	C₂×#486333, C₂×#485590, C₂×#486084, …
maximal quotient of	#72307464, #72307580

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391846);