C₄×#1791 #354785

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₄×#1791

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3622 conjugacy classes of subgroups; only the 582 normal subgroups are shown.

normal subgroups only

This group has 3622 conjugacy classes of subgroups — too many to draw. Only its 582 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1791, C₄×#21483, #72342950, …
maximal normal subgroup of	D₄×#1791, C₄×#21483, #72342950, …
maximal quotient of	#72343091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261981);