#184797

order	512
nilpotency class	3
isoclinism	isoclinic class of #184792
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

4062 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 4062 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,g^b=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314584
maximal normal subgroup of	#72314584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91993);