C₂×#7097 #122871

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7097

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

306 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 512

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30067);