C₂×#7448 #150948

order	512
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7448

Subgroups

	subgroup	G/H
center, socle	C₂²	#1361
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216259, C₂×#151634, #72216264, …
maximal normal subgroup of	#72216259, C₂×#151634, #72216264, …
maximal quotient of	#72225435, #72225474, #72227040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58144);