C₂×#12772 #493603

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#12772

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂²	#1790
prime radical	C₂×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2435 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2435 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261524, C₂×#511840, #72261616, …
maximal normal subgroup of	#72261524, C₂×#511840, #72261616, …
maximal quotient of	C₂×#149484

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400799);