C₂×#12175 #484656

order	512
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12175

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

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

normal subgroups only

This group has 912 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)(5,15)(6,19)(7,11)(8,13)(9,14)(10,16)(12,22)(17,31)(18,32)(20,25)(21,30)(23,29)(24,28)(26,34)(27,33);(3,5)(4,10)(6,20)(7,17)(8,24)(9,18)(11,25)(12,28)(13,26)(14,27)(15,19)(16,22)(21,34)(23,33)(29,31)(30,32);(4,11,14,13)(6,21)(10,25,27,26)(12,19,29,30)(15,31,32,28)(16,33)(20,34)(22,23)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486345, C₂×#486084, C₂×#485583, …
maximal normal subgroup of	C₂×#486345, C₂×#486084, C₂×#485583, …
maximal quotient of	#72265113, #72307580

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391852);