#481431

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	D₈×#41	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 240 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216824, #72216897, #72216933, …
maximal normal subgroup of	#72216824, #72216897, #72216933, …
maximal quotient of	#72285982, #72343711

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388627);