C₂²×#2614 #7627754

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂²×#2614

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

23774 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.

normal subgroups only

This group has 23774 conjugacy classes of subgroups — too many to draw. Only its 4258 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252228, C₂×#449480, C₂×#8093479, …
maximal normal subgroup of	#72252228, C₂×#449480, C₂×#8093479, …
maximal quotient of	C₂×#4308228, C₂²×#14873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534950);