C₄×#2875 #6369729

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄³ · Schur multiplier (H₂) = C₂⁶×C₄³

Aliases

direct product

C₄×#2875

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator	C₂	C₂²×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,3,4);(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(5,6,7,8)(10,14)(15,20)(16,21)(22,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72224762, #72224763, #72226779, …
maximal normal subgroup of	#72224762, #72224763, #72226779, …
maximal quotient of	#72224763, AGL(1,𝔽₅)×#2875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276925);

C4×#2875 #6369729