C₂×#7395 #150168

order	512
nilpotency class	5
isoclinism	isoclinic class of #7394
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7395

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	#381	C₂³
prime radical	C₂×#534	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵ ◁ C₂×#381 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

743 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 74 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);(3,4,7,11,8,12,20,28)(5,13,15,29,16,25,32,34)(6,17,18,33,19,10,24,26)(9,21,22,14,23,30,27,31);(3,5)(4,9)(6,14)(7,15)(8,16)(10,25)(11,27)(12,23)(13,17)(18,21)(19,31)(20,32)(22,28)(24,30)(26,34)(29,33)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#150395, #72285142, #72285169, …
maximal normal subgroup of	C₂×#150395, #72285142, #72285169, …
maximal quotient of	#72285142

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57364);