C₂²×#1397 #512540

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1397

Subgroups

	subgroup	G/H
center	C₂³×C₁₆	C₂²
commutator	C₂	C₂²×C₄×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle, prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2);(3,4);(9,10,11,16,12,17,19,28,13,18,20,29,22,30,31,38,14,15,21,25,23,26,32,35,24,27,33,36,34,37,39,40);(5,6,7,8)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40)

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#7330, #72231859, #72231862, …
maximal normal subgroup of	C₂²×#7330, #72231859, #72231862, …
maximal quotient of	#72321329

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419736);

C22×#1397 #512540