C₂×#7317 #128025

order	512
nilpotency class	4
isoclinism	isoclinic class of #7316
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7317

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

1304 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 146 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);(3,4,6,10,8,12,19,27)(5,13,14,25,16,9,17,23)(7,20,21,26,22,11,18,28)(15,24,31,34,32,29,30,33);(3,5,7,15,8,16,22,32)(4,9,11,24,12,13,20,29)(6,17,18,31,19,14,21,30)(10,25,26,34,27,23,28,33)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210062, #72210063, #72210064, …
maximal normal subgroup of	#72210062, #72210063, #72210064, …
maximal quotient of	C₂²×#7317, #72266598, C₄×#7317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35221);