#105416

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1016 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238897
maximal normal subgroup of	#72238897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12612);