C₄×#3407 #7622418

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₄×#3407

Subgroups

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

2444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₈×#3407, #72227038, C₄×#33329, …
maximal normal subgroup of	C₈×#3407, #72227038, C₄×#33329, …
maximal quotient of	C₈×#3407, M₃₂×#131, #72227038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529614);

C4×#3407 #7622418