C₂³×M₆₄ #7625180

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₆ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂³×M₆₄

Subgroups

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

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 38 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1398, C₂³×#2263, C₂³×#1419, …
maximal normal subgroup of	C₂³×#1398, C₂³×#2263, C₂³×#1419, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532376);

C23×M64 #7625180