C₂³×#504 #10585845

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₈ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂³×#504

Subgroups

	subgroup	G/H
center	C₂³×C₁₆	C₂²
commutator	C₂	C₂⁵×C₈
Frattini, cosocle	C₈	C₂⁶
socle	C₂⁴	C₂²×C₈
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

8162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 38 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#2171, C₂×D₄×#504
maximal normal subgroup of	C₂³×#2171, C₂×D₄×#504
maximal quotient of	C₂×D₄×#504, #72322106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493041);

C23×#504 #10585845