C₄×#1398 #151749

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₄×#1398

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 3 generators on 36 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#1398, #72216305, #72216315, …
maximal normal subgroup of	C₈×#1398, #72216305, #72216315, …
maximal quotient of	C₈×#1398, C₂×C₄×#1398, #72225138, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58945);