C₂²×#1321 #480006

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1321

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂³	#323
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1618 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1618 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2);(3,4);(5,6,9,14,10,13,24,30,11,15,25,32,8,20,22,31)(7,16,18,33,17,12,26,27,19,29,23,35,21,28,34,36);(5,7)(6,12)(8,21)(9,23)(10,17)(11,19)(13,29)(14,27)(15,28)(16,20)(18,25)(22,26)(24,34)(30,35)(31,33)(32,36)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213777, #72215049, #72216369, …
maximal normal subgroup of	#72213777, #72215049, #72216369, …
maximal quotient of	C₂²×#7105

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387202);