#94296

order	512
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1351
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)(7,15,17,29)(8,12)(9,13)(10,14)(11,16,23,19)(18,34,33,30)(20,27)(21,26)(22,28)(24,32,36,35)(25,31);(5,7,9,18,10,17,22,33)(6,11,13,24,14,23,28,36)(8,19,20,35,21,16,31,32)(12,15,25,30,26,29,27,34)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,e^2=i,b^a=bc,c^a=cf,e^a=efg,h^a=hi,c^b=cg,d^b=di,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, #72267758, #72285374
maximal normal subgroup of	#72238062, #72267758, #72285374
maximal quotient of	#72238068

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1492);