#94292

order	512
nilpotency class	5
isoclinism	isoclinic class of #7394
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₂²⋊C₄	C₄²
Frattini, cosocle	C₂×#381	C₂²
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵ ◁ C₂×#381 ◁ G
LECS	G ▷ C₂×#381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,e^2=i,b^a=bc,c^a=cf,e^a=efgi,h^a=hi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238062, #72267777, #72285374
maximal normal subgroup of	#72238062, #72267777, #72285374
maximal quotient of	#72238067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1488);