#154190

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

4118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328480, #72329023, #72329061
maximal normal subgroup of	#72328480, #72329023, #72329061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61386);